Oligopoly Models
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Oligopoly Models Patrick Bajari Econ 4631
Patrick Bajari Econ 4631 ()
Oligopoly Models
1 / 55
Motivation
In chapter 6 we will discuss game theoretic models of competition In many markets, there are a small number of dominant …rms that interact Examples: 1
National TV Networks- NBC, ABC, CBS, Fox
2
Automobiles- Toyota, Ford, GM, Honda,
3
Micro Processors- Intel and AMD
4
Upper Midwest Research Universities- Minnesota, Wisconsin, Michigan, Michigan State, Chicago, Northwestern, Urbana, Iowa, Iowa Stata
Patrick Bajari Econ 4631 ()
Oligopoly Models
2 / 55
Motivation
When making prices, output or other strategic decisions, these …rms know that their decisions are interrelated This violates a major assumption of the basic competitive model In that model, …rms are small and prices are taken as given How do we analyze these games?
Patrick Bajari Econ 4631 ()
Oligopoly Models
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Nash Equilibrium
There are i = 1, ..., N players in the game Let ai be a strategy for player i Let π i (ai , a i ) be the payo¤s of player i as function of i’s strategy ai and the strategy a i of all the other players
Patrick Bajari Econ 4631 ()
Oligopoly Models
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Nash Equilibrium An equilibrium a1 , ..., aN is a collection of strategies that are maximizing, holding the strategies of the other agents …xed That is, for all i ai maximizes π i (ai , a i ) In words, Nash equilibrium has two assumptions: 1
Maximization- that is, agents choose the strategy that is in their best interests
2
Rational Expectations- You anticipate other players equilibrium actions a i
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Oligopoly Models
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Example
Battle of the sexes Pat/Pat’s wife Ashton Kutcher Movie Ice Fishing
Patrick Bajari Econ 4631 ()
Ashton Kutcher Movie (1,3) (0,0)
Oligopoly Models
Ice Fishing (0,0) (3,1)
6 / 55
Prisoner’s Dilemma
Fink/Don’t Fink Each player serves 1 year if they don’t …nk If one player …nks, he gets no penalties while the player that doesn’t gets 10 years If both players cooperate, they both get 3 years 1/2 Don’t Fink Fink
Don’t Fink (-.5,-.5) (0,-10)
Patrick Bajari Econ 4631 ()
Fink (-10,0) (-3,-3)
Oligopoly Models
7 / 55
Rationality assumptions
Is Nash equilibrium a reasonable assumption? Human stupidity is certainly present in some market situations Why do we use models of rationality?
Patrick Bajari Econ 4631 ()
Oligopoly Models
8 / 55
Rationality assumptions
1
In some markets, people are smart and they make decisions well. If they can’t make a decision well, they will hire someone or acquire someone to assist them with a decision Calculus is a metaphor for optimization Just because market participants can’t solve our models, doesn’t mean they aren’t useful
Patrick Bajari Econ 4631 ()
Oligopoly Models
9 / 55
Rationality assumptions
2. Rationality is the outcome of simpler and less demanding forms of reinforcement. Maynard Smith has noted that Nash equilibrium can be the outcome of evolution He imagines strategies replicating based on their success, analogous to sexual reproduction
Patrick Bajari Econ 4631 ()
Oligopoly Models
10 / 55
Rationality assumptions
Sexual reproduction does not require an understanding of calculus The reason why it leads to equilibrium is pretty clear. Since there is "survival of the …ttest", strategies with poor payo¤s will be weeded out If the dynamics of survival of the …ttest settle down, it must be an equilibrium Otherwise, there is always a "mutant" strategy that can pro…tably enter with higher payo¤ and have proportionally more o¤spring Is a trout rational or a ‡y? Probably not, but it acts as if it was rational.
Patrick Bajari Econ 4631 ()
Oligopoly Models
11 / 55
Rationality Assumptions
In evolutionary model, strategies with higher payo¤s survive because these players "die" and are less likely to reproduce Markets can have strong reinforcement e¤ects as well Instead of dying, you go broke Firms and other economic actors are often punished if they make decisions that lead to lower pro…tability
Patrick Bajari Econ 4631 ()
Oligopoly Models
12 / 55
Rationality Assumptions
Just like in our evolutionary models, if bad strategies are punished, the only thing that you can converge to is equilibrium Economists have studied "evolutionary models" To be candid, the academic economics literature has not fully appreciated the power of Darwin’s ideas Learning is another adjustment process For example, you maximize forming your beliefs given what you have seen in the past
Patrick Bajari Econ 4631 ()
Oligopoly Models
13 / 55
Rationality Assumptions
Just like in our evolutionary models, if bad strategies are punished, the only thing that you can converge to is equilibrium Economists have studied "evolutionary models" To be candid, the academic economics literature has not fully appreciated the power of Darwin’s ideas Learning is another adjustment process For example, you maximize forming your beliefs given what you have seen in the past
Patrick Bajari Econ 4631 ()
Oligopoly Models
14 / 55
Rationality Assumptions
4. It can be hard to beat the market A lack of rationality implies that an academic economist can make a better decision than a …rm in the marketplace In general, this is much harder than it seems from the outside
Patrick Bajari Econ 4631 ()
Oligopoly Models
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Rationality Assumptions 5. The economic models of irrationality just aren’t that good Much of the work in behavioral economics documents various "anamolies" in decision making It is often disputed whether the anamolies actually exist or are a function of measurement problems (e.g. biased forecasts by analysts) There is no general and logically coherent framework that can be applied as broadly as Nash equilibrium models Economics is distinguished by having formal and logically coherent analysis based on math While there are competing traditions in economics, they have not made nearly the progress in the past 100 years
Patrick Bajari Econ 4631 ()
Oligopoly Models
16 / 55
Rationality Assumptions
None of this is meant to say that we won’t eventually …gure out how to build better models that perturb the rationality assumptions Neuro economics seems to be doing some quite precise measurement and interesting work Some behavioral work in experimental economics is also quite good
Patrick Bajari Econ 4631 ()
Oligopoly Models
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Cournot
In the Cournot model, N …rms compete by producing a homogenous good The strategy of each …rm is qi the amount of good i to produce Let the market quantity Q be de…ned by: Q = q1 + ... + qN
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Oligopoly Models
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Cournot
Let P (Q ) denote the inverse demand function, that is, the price as a function of the aggregate quantity Q Let c denote the unit cost to …rm i of producing the good For simplicity assume constant marginal costs
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Oligopoly Models
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Cournot
Then the pro…ts of …rm i can be written as: π i (qi , q i ) = (P (q1 + ... + qN )
c ) qi
A Nash equilibrium is a vector of quantities q1 , ..., qN such that: qi maximizes π i (qi , q i ) Let’s study the properties of the maximization problem
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Oligopoly Models
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Residual Demand
It is useful to talk about the "residual demand curve" When i solves her maximization problem, she takes Q where Q i = q1 + ... + qi 1 + qi + ... + qN
i
as given,
It is useful to write the equilibrium price as P (qi + Q i ) We call this the residual demand (or more properly, residual inverse demand), because you express the price as a function of i’s quantity, holding every one else’s quantity …xed
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Oligopoly Models
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Residual Demand For illustration, assume that P (qi + Q i ) is linear, i.e. p=a
b (qi + Q i )
Note that the elasticity of the residual demand can be written as: %∆qi %∆p
= = =
dqi p dp qi 1 a b (qi + Q i ) b qi Q i a b bqi qi
Thus residual demand becomes more elastic as Q i increases A percentage increase in price requires an larger percentage decrease in quantity If Q i is near in…nity, this means that a 1 percent increase in price would require a nearly in…nite percentage increase in quantity Patrick Bajari Econ 4631 ()
Oligopoly Models
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Residual Demand
Thus, even if the aggregate demand curve P (Q ) is quite inelastic, it may be the case that residual demand in very elastic From the perspective of the …rm, the important thing in determining output is the elasticity of residual demand, not aggregate demand!
Patrick Bajari Econ 4631 ()
Oligopoly Models
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Equilibrium First order conditions: π i (qi , q i ) = (P (qi + Q i ) c ) qi d dP qi + P c π i (qi , q i ) = dqi dQ At a maximum, the derivative must be zero: dP qi dQ
P
c
= 0
P
c
=
P
c P
Patrick Bajari Econ 4631 ()
=
Oligopoly Models
dP qi dQ dP qi dQ P
24 / 55
Equilibrium Let assume that the …rms choose a symmetric strategy That is, since they all have the same cost, they all produce the same output Then qi = q =
Q N
and hence
P
c P
= = =
dP Q 1 dQ P N 1 dQ P dP Q N 1 εN
where ε is the market elasticity of demand Patrick Bajari Econ 4631 ()
Oligopoly Models
25 / 55
Comparative Statics
Our model implies that markups increase as ε gets more inelastic This seems reasonable- the less abilty consumers have to substitute, the higher the price cost margin or Lerner index P P c should be The model also implies that as N becomes larger, the price cost margins decrease This also seems reasonable intuitively- as there is more competition, …rms have less market power
! ∞, we get that prices converge to marginal cost That is, Cournot converges to perfect competition If we send N
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Oligopoly Models
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Hirschman-Her…ndal Index
Next we wish to derive the Hirschman-Her…ndal Index or HHIH This is a famous index used to measure market power and is used in the merger guidelines The HHI is motivated in part by the Cournot model
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Oligopoly Models
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Hirschman-Her…ndal Index Suppose that we have asymmetric …rms, i.e. the marginal costs are ci which need not be equal across …rms The …rst order conditions in this case become: π i (qi , q i ) = (P (qi + Q i ) ci ) qi d dP qi + P ci π i (qi , q i ) = dqi dQ At a maximum, the derivative must be zero: dP qi dQ
P
ci
P
ci P
= 0 = =
dP Q qi dQ P Q si ε
Where ε is the elasticity of demand for …rm i and si is i’s share Patrick Bajari Econ 4631 ()
Oligopoly Models
28 / 55
Hirschman-Her…ndal Index The HHI is the market share weighted price-cost or Lerner index (when the elasticity of demand is one): N
HHI
=
∑
P
i =1 N
=
ci P
1 ε i∑ =1
si
si2
The HHI tells us that the square of market shares is related to price cost margins (holding aggregate demand elasticity constant) The bigger
P ci P
the larger the ine¢ ciency from market power
Hence, the use of HHI in merger analysis
Patrick Bajari Econ 4631 ()
Oligopoly Models
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Hirschman-Her…ndal Index
Note that the HHI only measures market power under the assumptions of the Cournot model If the market involves di¤erentiated products, then the HHI is a misleading measure This was certainly the case in Wild Oats/Whole Foods Their share was small (depending on how you de…ned the market), but pricing pressures and competitive e¤ects could have been large That is why antitrust guidelines may move towards HHI for homogenous goods markets and diversion ratios (discussed below) for di¤erentiated product markets
Patrick Bajari Econ 4631 ()
Oligopoly Models
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Di¤erentiated product market
Many of the markets that we encounter in economics are di¤erentiated product markets That is, …rms sell similar, but not identical goods Examples: Autos, Cell Phones, Computers, Universities In di¤erentiated product models, we usually model competition as Bertrand That is, …rms set prices as their strategic variable
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The Hedonic Approach
In di¤erentiated product markets, it is typical to model consumer behavior using a hedonic approach In this approach, we model a good as a bundle of characteristics For example, a laptop could be viewed as a bundle of: 1
Screen size
2
CPU speed
3
Weight
4
Brand (e.g. PC or Mac)
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Oligopoly Models
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The Hedonic Approach Suppose that there are j = 1, ..., J goods and i = 1, ..., N consumers in a market Let uij denote the utility of consumer i for good j A classic empirical model of demand is due to McFadden and is called the conditional logit (he won the Nobel prize in large part for his work related to this model) We write uij as: uij = β0 + screenj β1 + CPUj β2 + Weightj β3
αpj + εij
In the above, a consumer’s utility for good j is a function of its characteristics. The parameters β represent the "marginal utility" for these characteristics α represents the disutility of paying a higher price (for the formally inclined, this is actually an indirect utility function) εij is an iid shock to agents preferences Patrick Bajari Econ 4631 ()
Oligopoly Models
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The Hedonic Approach Suppose that there are j = 1, ..., J goods and i = 1, ..., N consumers in a market Here consumers make discreet choices, e.g. choices between mutually exclusive goods Let uij denote the utility of consumer i for good j A classic empirical model of demand is due to McFadden and is called the conditional logit (he won the Nobel prize in large part for his work related to this model) We write uij as: uij = β0 + screenj β1 + CPUj β2 + Weightj β3
αpj + εij
Households are assumed to be utility maximizers, e.g. i chooses j if and only if uij > uij 0 for all j 0 6= i Patrick Bajari Econ 4631 ()
Oligopoly Models
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The Hedonic Approach
In the above, a consumer’s utility for good j is a function of its characteristics. The parameters β represent the "marginal utility" for these characteristics α represents the disutility of paying a higher price (for the formally inclined, this is actually an indirect utility function) εij is a random iid shock to agents preferences
Patrick Bajari Econ 4631 ()
Oligopoly Models
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The Hedonic Approach
The random preference shock captures preference heterogeneity This is very important in understanding these markets People have very distinct preferences even after conditioning on obvious demographic variables: (e.g. location, age, income, etc...) We use econometrics to measure β- actually, it’s a pretty simple example of maximum likelihood estimation
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The Conditional Logit
McFadden noted that if you let εij be an "extreme value" random variable, we get some very pretty formulas In particular, the probabiltiy that i chooses good j is equal to: Pr(i chooses j ) = exp ( β0 + screenj β1 + CPUj β2 + Weightj β3 αpj ) ∑j 0 exp ( β0 + screenj 0 β1 + CPUj 0 β2 + Weightj 0 β3 αpj 0 ) Note that as pj " the probability that j is chosen goes down
As desirable characteristics are added, the probability that j is chosen goes up
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Oligopoly Models
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The Conditional Logit
The market share and total quantity of j is equal to: sj (pj , p j ) = exp ( β0 + screenj β1 + CPUj β2 + Weightj β3 αpj ) ∑j 0 exp ( β0 + screenj 0 β1 + CPUj 0 β2 + Weightj 0 β3 αpj 0 ) Qj = N
sj (pj , p j )
It can be veri…ed that the elasticity of the residual demand curve facing j is: %∆sj = αpj sj %∆pj
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Bertrand Equilibrium A Bertrand equilibrium is a Nash equilibrium in prices The pro…t of j is:
(pj
cj )Nsj (pj , p j )
The …rst order condtions are: ∂sj sj + (pj cj ) ∂pj
= 0
(pj
cj ) =
(pj
cj ) pj
= =
∆pj ∆sj sj ∆pj = pj ∆sj 1 εj
sj
%∆pj %∆sj
Where εj is the residual demand elasticity for good j Patrick Bajari Econ 4631 ()
Oligopoly Models
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Measuring Market Power
The formula below is called the inverse elasticity rule:
(pj
cj ) pj
=
1 εj
This rule states that price cost margins are higher (and market power generates more ine¢ ciency), if the residual demand is more inelastic IO economists have been obsessed with the measurement of market power For example, they will go to the data and attempt to estimate εj
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Oligopoly Models
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Measuring Market Power
In the logit model, for example, εj =
αsj pj
If we know α, we can infer the Lerner index This is useful in practice since we can identify the most distorted markets Also, we can perhaps identify markets in which mergers might be particularly bad or compute the e¤ects of collusion This all boils down to determining how mergers or collusion changed residual demand elasticities
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Oligopoly Models
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Calibration of the logit model
It is possible for us to calibrate the logit model in order to ask some interesting economic questions As an example, let’s consider the e¤ects of the proposed Google Yahoo! ad deal that was blocked by the Department of Justice Google- 65% Bing-10% Yahoo!-25%
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Calibration of the logit model
Search engines make money by selling advertisements Advertisers bid for "sponsored links" on these sites They pay their bid times each click that is generated by a search
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Oligopoly Models
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Calibration of the logit model
Suppose that the utility of search engine j takes the form: uij = δj
αpj + εij
Where δj is the "mean" utility that an advertiser has for search engine j pj is the advertising price of search engine j εij is the preference shock for engine j
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Calibration of the logit model To simplify things, let’s suppose that prices of a click are equalized across sites Normalize prices to $1. Assume that Google’s price cost margin is 27% (this is from accounting data, not economic costs) This implies that:
p
c p
=
.27 = α =
Patrick Bajari Econ 4631 ()
1 αsj pj 1 α.65 1 = 5.6980 .27 .65 Oligopoly Models
45 / 55
Calibration of the logit model
Next, let’s solve for the δj sGoog sBing sYahoo D
exp(δGoog 5.6980) D exp(δBing 5.6980) = .1 = D exp(δYahoo 5.6980) = .25 = D = exp(δGoog 5.6980) + exp(δBing
= .65 =
+ exp(δYahoo
5.6980)
5.6980)
Since we know all the shares, this is 3 equations in 3 unknowns
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Calibration of the logit model
Without loss of generality, we can normalize δBing = 0 Verify this by adding a constant to all of the δj Then we can solve for δGoog and δYahoo by: sGoog sBing δGoog δYahoo
Patrick Bajari Econ 4631 ()
.65 exp(δGoog 5.6980) .1 exp( 5.6980) = log(.65) log(.1) = 1.8718
=
= log(.25)
Oligopoly Models
log(.1) = .9163
47 / 55
Post Merger Outcome
If we remove Yahoo! from the market, the shares of Bing and Google are: sGoog
=
sGoog
=
exp(1.8718 5.6980pGoog ) exp(1.8718 5.6980pGoog ) + exp( 5.6980pBing ) exp( 5.6980pBing ) exp(1.8718 5.6980pGoog ) + exp( 5.6980pBing )
Let’s give both Bing and Google costs of .83 cents (let’s ad in 10 cents of "economic costs" to the price cost margins)
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Post Merger Outcome
In a Nash equilibrium, the …rms would maximize:
(pGoog (pBing
exp(1.8718 5.6980pGoog ) exp(1.8718 5.6980pGoog ) + exp( 5.6980pBing ) exp( 5.6980pBing ) .83) exp(1.8718 5.6980pGoog ) + exp( 5.6980pBing ) .83)
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Post Merger Outcome
The Nash equilibrium would be the following two equations in the two unknowns pGoog , pBing
(pGoog .83) 1 = exp ( 1.8718 5.6980p ) pGoog 5.6980 exp (1.8718 5.6980p )+expGoog ( 5. Goog (pBing .83) pBing
=
1 exp ( 5.6980p Bing ) 5.6980 exp (1.8718 5.6980p )+exp p ( 5.6980p Bing ) Bing Goog
This can’t be solved for analytically. Instead, we do it numerically on the computer
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Oligopoly Models
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Calibration of the logit model
The post merger prices are pGoog = 1.4509, pBing = 1.2091 This is a very large price increase relative to the baseline prices of 1 This back of the envelope calculation suggests this acquisition could have been very harmful to advertisers
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Extensions of the Logit
Remark: What happens after this point of the lecture is a bit advanced and will not be on the exam While the logit model is widely used, there has been important work on extending this model In particular, researchers have found it important to include "random coe¢ cients in the logit" In the logit, the only reason why di¤erent people make di¤erent choices is because they draw di¤erent εij In the random coe¢ cient model, di¤erent people have di¤erent marginal utilities In practice, this turns out to be a very important extension
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Extensions of the Logit You will not be tested on this section One formulation of this model is to suppose that there are r = 1, ..., R types of consumers These types have utilities characterized by the parameters β(r ) , α(r ) This leads to the utility function (r )
(r )
(r )
(r )
uij = β0 + screenj β1 + CPUj β2 + Weightj β3
α(r ) pj + εij
Households are assumed to be utility maximizers, e.g. i chooses j if and only if uij > uij 0 for all j 0 6= i The probability that type r chooses good j is:
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Extensions of the Logit The type r’s choice probability is:
exp
(r ) β0
Pr(i chooses j jr ) = (r )
(r )
α(r ) pj
+ screenj β1 + CPUj β2
(r )
(r )
(r )
∑j 0 exp β0 + screenj 0 β1 + CPUj 0 β2
α(r ) pj 0
Let p (r ) denote the probability of type r The choice probability for j is found by summing over r
Pr( j chosen) = (r )
R
∑
r =1
p (r )
(r )
(r )
exp β0 + screenj β1 + CPUj β2 (r )
(r )
(r )
∑j 0 exp β0 + screenj 0 β1 + CPUj 0 β2
Patrick Bajari Econ 4631 ()
Oligopoly Models
α(r ) pj α(r ) pj 0 54 / 55
Extensions of the Logit
Next lecture, we will use this framework to study music piracy
Patrick Bajari Econ 4631 ()
Oligopoly Models
55 / 55
Patrick Bajari Econ 4631 ()
Oligopoly Models
1 / 55
Motivation
In chapter 6 we will discuss game theoretic models of competition In many markets, there are a small number of dominant …rms that interact Examples: 1
National TV Networks- NBC, ABC, CBS, Fox
2
Automobiles- Toyota, Ford, GM, Honda,
3
Micro Processors- Intel and AMD
4
Upper Midwest Research Universities- Minnesota, Wisconsin, Michigan, Michigan State, Chicago, Northwestern, Urbana, Iowa, Iowa Stata
Patrick Bajari Econ 4631 ()
Oligopoly Models
2 / 55
Motivation
When making prices, output or other strategic decisions, these …rms know that their decisions are interrelated This violates a major assumption of the basic competitive model In that model, …rms are small and prices are taken as given How do we analyze these games?
Patrick Bajari Econ 4631 ()
Oligopoly Models
3 / 55
Nash Equilibrium
There are i = 1, ..., N players in the game Let ai be a strategy for player i Let π i (ai , a i ) be the payo¤s of player i as function of i’s strategy ai and the strategy a i of all the other players
Patrick Bajari Econ 4631 ()
Oligopoly Models
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Nash Equilibrium An equilibrium a1 , ..., aN is a collection of strategies that are maximizing, holding the strategies of the other agents …xed That is, for all i ai maximizes π i (ai , a i ) In words, Nash equilibrium has two assumptions: 1
Maximization- that is, agents choose the strategy that is in their best interests
2
Rational Expectations- You anticipate other players equilibrium actions a i
Patrick Bajari Econ 4631 ()
Oligopoly Models
5 / 55
Example
Battle of the sexes Pat/Pat’s wife Ashton Kutcher Movie Ice Fishing
Patrick Bajari Econ 4631 ()
Ashton Kutcher Movie (1,3) (0,0)
Oligopoly Models
Ice Fishing (0,0) (3,1)
6 / 55
Prisoner’s Dilemma
Fink/Don’t Fink Each player serves 1 year if they don’t …nk If one player …nks, he gets no penalties while the player that doesn’t gets 10 years If both players cooperate, they both get 3 years 1/2 Don’t Fink Fink
Don’t Fink (-.5,-.5) (0,-10)
Patrick Bajari Econ 4631 ()
Fink (-10,0) (-3,-3)
Oligopoly Models
7 / 55
Rationality assumptions
Is Nash equilibrium a reasonable assumption? Human stupidity is certainly present in some market situations Why do we use models of rationality?
Patrick Bajari Econ 4631 ()
Oligopoly Models
8 / 55
Rationality assumptions
1
In some markets, people are smart and they make decisions well. If they can’t make a decision well, they will hire someone or acquire someone to assist them with a decision Calculus is a metaphor for optimization Just because market participants can’t solve our models, doesn’t mean they aren’t useful
Patrick Bajari Econ 4631 ()
Oligopoly Models
9 / 55
Rationality assumptions
2. Rationality is the outcome of simpler and less demanding forms of reinforcement. Maynard Smith has noted that Nash equilibrium can be the outcome of evolution He imagines strategies replicating based on their success, analogous to sexual reproduction
Patrick Bajari Econ 4631 ()
Oligopoly Models
10 / 55
Rationality assumptions
Sexual reproduction does not require an understanding of calculus The reason why it leads to equilibrium is pretty clear. Since there is "survival of the …ttest", strategies with poor payo¤s will be weeded out If the dynamics of survival of the …ttest settle down, it must be an equilibrium Otherwise, there is always a "mutant" strategy that can pro…tably enter with higher payo¤ and have proportionally more o¤spring Is a trout rational or a ‡y? Probably not, but it acts as if it was rational.
Patrick Bajari Econ 4631 ()
Oligopoly Models
11 / 55
Rationality Assumptions
In evolutionary model, strategies with higher payo¤s survive because these players "die" and are less likely to reproduce Markets can have strong reinforcement e¤ects as well Instead of dying, you go broke Firms and other economic actors are often punished if they make decisions that lead to lower pro…tability
Patrick Bajari Econ 4631 ()
Oligopoly Models
12 / 55
Rationality Assumptions
Just like in our evolutionary models, if bad strategies are punished, the only thing that you can converge to is equilibrium Economists have studied "evolutionary models" To be candid, the academic economics literature has not fully appreciated the power of Darwin’s ideas Learning is another adjustment process For example, you maximize forming your beliefs given what you have seen in the past
Patrick Bajari Econ 4631 ()
Oligopoly Models
13 / 55
Rationality Assumptions
Just like in our evolutionary models, if bad strategies are punished, the only thing that you can converge to is equilibrium Economists have studied "evolutionary models" To be candid, the academic economics literature has not fully appreciated the power of Darwin’s ideas Learning is another adjustment process For example, you maximize forming your beliefs given what you have seen in the past
Patrick Bajari Econ 4631 ()
Oligopoly Models
14 / 55
Rationality Assumptions
4. It can be hard to beat the market A lack of rationality implies that an academic economist can make a better decision than a …rm in the marketplace In general, this is much harder than it seems from the outside
Patrick Bajari Econ 4631 ()
Oligopoly Models
15 / 55
Rationality Assumptions 5. The economic models of irrationality just aren’t that good Much of the work in behavioral economics documents various "anamolies" in decision making It is often disputed whether the anamolies actually exist or are a function of measurement problems (e.g. biased forecasts by analysts) There is no general and logically coherent framework that can be applied as broadly as Nash equilibrium models Economics is distinguished by having formal and logically coherent analysis based on math While there are competing traditions in economics, they have not made nearly the progress in the past 100 years
Patrick Bajari Econ 4631 ()
Oligopoly Models
16 / 55
Rationality Assumptions
None of this is meant to say that we won’t eventually …gure out how to build better models that perturb the rationality assumptions Neuro economics seems to be doing some quite precise measurement and interesting work Some behavioral work in experimental economics is also quite good
Patrick Bajari Econ 4631 ()
Oligopoly Models
17 / 55
Cournot
In the Cournot model, N …rms compete by producing a homogenous good The strategy of each …rm is qi the amount of good i to produce Let the market quantity Q be de…ned by: Q = q1 + ... + qN
Patrick Bajari Econ 4631 ()
Oligopoly Models
18 / 55
Cournot
Let P (Q ) denote the inverse demand function, that is, the price as a function of the aggregate quantity Q Let c denote the unit cost to …rm i of producing the good For simplicity assume constant marginal costs
Patrick Bajari Econ 4631 ()
Oligopoly Models
19 / 55
Cournot
Then the pro…ts of …rm i can be written as: π i (qi , q i ) = (P (q1 + ... + qN )
c ) qi
A Nash equilibrium is a vector of quantities q1 , ..., qN such that: qi maximizes π i (qi , q i ) Let’s study the properties of the maximization problem
Patrick Bajari Econ 4631 ()
Oligopoly Models
20 / 55
Residual Demand
It is useful to talk about the "residual demand curve" When i solves her maximization problem, she takes Q where Q i = q1 + ... + qi 1 + qi + ... + qN
i
as given,
It is useful to write the equilibrium price as P (qi + Q i ) We call this the residual demand (or more properly, residual inverse demand), because you express the price as a function of i’s quantity, holding every one else’s quantity …xed
Patrick Bajari Econ 4631 ()
Oligopoly Models
21 / 55
Residual Demand For illustration, assume that P (qi + Q i ) is linear, i.e. p=a
b (qi + Q i )
Note that the elasticity of the residual demand can be written as: %∆qi %∆p
= = =
dqi p dp qi 1 a b (qi + Q i ) b qi Q i a b bqi qi
Thus residual demand becomes more elastic as Q i increases A percentage increase in price requires an larger percentage decrease in quantity If Q i is near in…nity, this means that a 1 percent increase in price would require a nearly in…nite percentage increase in quantity Patrick Bajari Econ 4631 ()
Oligopoly Models
22 / 55
Residual Demand
Thus, even if the aggregate demand curve P (Q ) is quite inelastic, it may be the case that residual demand in very elastic From the perspective of the …rm, the important thing in determining output is the elasticity of residual demand, not aggregate demand!
Patrick Bajari Econ 4631 ()
Oligopoly Models
23 / 55
Equilibrium First order conditions: π i (qi , q i ) = (P (qi + Q i ) c ) qi d dP qi + P c π i (qi , q i ) = dqi dQ At a maximum, the derivative must be zero: dP qi dQ
P
c
= 0
P
c
=
P
c P
Patrick Bajari Econ 4631 ()
=
Oligopoly Models
dP qi dQ dP qi dQ P
24 / 55
Equilibrium Let assume that the …rms choose a symmetric strategy That is, since they all have the same cost, they all produce the same output Then qi = q =
Q N
and hence
P
c P
= = =
dP Q 1 dQ P N 1 dQ P dP Q N 1 εN
where ε is the market elasticity of demand Patrick Bajari Econ 4631 ()
Oligopoly Models
25 / 55
Comparative Statics
Our model implies that markups increase as ε gets more inelastic This seems reasonable- the less abilty consumers have to substitute, the higher the price cost margin or Lerner index P P c should be The model also implies that as N becomes larger, the price cost margins decrease This also seems reasonable intuitively- as there is more competition, …rms have less market power
! ∞, we get that prices converge to marginal cost That is, Cournot converges to perfect competition If we send N
Patrick Bajari Econ 4631 ()
Oligopoly Models
26 / 55
Hirschman-Her…ndal Index
Next we wish to derive the Hirschman-Her…ndal Index or HHIH This is a famous index used to measure market power and is used in the merger guidelines The HHI is motivated in part by the Cournot model
Patrick Bajari Econ 4631 ()
Oligopoly Models
27 / 55
Hirschman-Her…ndal Index Suppose that we have asymmetric …rms, i.e. the marginal costs are ci which need not be equal across …rms The …rst order conditions in this case become: π i (qi , q i ) = (P (qi + Q i ) ci ) qi d dP qi + P ci π i (qi , q i ) = dqi dQ At a maximum, the derivative must be zero: dP qi dQ
P
ci
P
ci P
= 0 = =
dP Q qi dQ P Q si ε
Where ε is the elasticity of demand for …rm i and si is i’s share Patrick Bajari Econ 4631 ()
Oligopoly Models
28 / 55
Hirschman-Her…ndal Index The HHI is the market share weighted price-cost or Lerner index (when the elasticity of demand is one): N
HHI
=
∑
P
i =1 N
=
ci P
1 ε i∑ =1
si
si2
The HHI tells us that the square of market shares is related to price cost margins (holding aggregate demand elasticity constant) The bigger
P ci P
the larger the ine¢ ciency from market power
Hence, the use of HHI in merger analysis
Patrick Bajari Econ 4631 ()
Oligopoly Models
29 / 55
Hirschman-Her…ndal Index
Note that the HHI only measures market power under the assumptions of the Cournot model If the market involves di¤erentiated products, then the HHI is a misleading measure This was certainly the case in Wild Oats/Whole Foods Their share was small (depending on how you de…ned the market), but pricing pressures and competitive e¤ects could have been large That is why antitrust guidelines may move towards HHI for homogenous goods markets and diversion ratios (discussed below) for di¤erentiated product markets
Patrick Bajari Econ 4631 ()
Oligopoly Models
30 / 55
Di¤erentiated product market
Many of the markets that we encounter in economics are di¤erentiated product markets That is, …rms sell similar, but not identical goods Examples: Autos, Cell Phones, Computers, Universities In di¤erentiated product models, we usually model competition as Bertrand That is, …rms set prices as their strategic variable
Patrick Bajari Econ 4631 ()
Oligopoly Models
31 / 55
The Hedonic Approach
In di¤erentiated product markets, it is typical to model consumer behavior using a hedonic approach In this approach, we model a good as a bundle of characteristics For example, a laptop could be viewed as a bundle of: 1
Screen size
2
CPU speed
3
Weight
4
Brand (e.g. PC or Mac)
Patrick Bajari Econ 4631 ()
Oligopoly Models
32 / 55
The Hedonic Approach Suppose that there are j = 1, ..., J goods and i = 1, ..., N consumers in a market Let uij denote the utility of consumer i for good j A classic empirical model of demand is due to McFadden and is called the conditional logit (he won the Nobel prize in large part for his work related to this model) We write uij as: uij = β0 + screenj β1 + CPUj β2 + Weightj β3
αpj + εij
In the above, a consumer’s utility for good j is a function of its characteristics. The parameters β represent the "marginal utility" for these characteristics α represents the disutility of paying a higher price (for the formally inclined, this is actually an indirect utility function) εij is an iid shock to agents preferences Patrick Bajari Econ 4631 ()
Oligopoly Models
33 / 55
The Hedonic Approach Suppose that there are j = 1, ..., J goods and i = 1, ..., N consumers in a market Here consumers make discreet choices, e.g. choices between mutually exclusive goods Let uij denote the utility of consumer i for good j A classic empirical model of demand is due to McFadden and is called the conditional logit (he won the Nobel prize in large part for his work related to this model) We write uij as: uij = β0 + screenj β1 + CPUj β2 + Weightj β3
αpj + εij
Households are assumed to be utility maximizers, e.g. i chooses j if and only if uij > uij 0 for all j 0 6= i Patrick Bajari Econ 4631 ()
Oligopoly Models
34 / 55
The Hedonic Approach
In the above, a consumer’s utility for good j is a function of its characteristics. The parameters β represent the "marginal utility" for these characteristics α represents the disutility of paying a higher price (for the formally inclined, this is actually an indirect utility function) εij is a random iid shock to agents preferences
Patrick Bajari Econ 4631 ()
Oligopoly Models
35 / 55
The Hedonic Approach
The random preference shock captures preference heterogeneity This is very important in understanding these markets People have very distinct preferences even after conditioning on obvious demographic variables: (e.g. location, age, income, etc...) We use econometrics to measure β- actually, it’s a pretty simple example of maximum likelihood estimation
Patrick Bajari Econ 4631 ()
Oligopoly Models
36 / 55
The Conditional Logit
McFadden noted that if you let εij be an "extreme value" random variable, we get some very pretty formulas In particular, the probabiltiy that i chooses good j is equal to: Pr(i chooses j ) = exp ( β0 + screenj β1 + CPUj β2 + Weightj β3 αpj ) ∑j 0 exp ( β0 + screenj 0 β1 + CPUj 0 β2 + Weightj 0 β3 αpj 0 ) Note that as pj " the probability that j is chosen goes down
As desirable characteristics are added, the probability that j is chosen goes up
Patrick Bajari Econ 4631 ()
Oligopoly Models
37 / 55
The Conditional Logit
The market share and total quantity of j is equal to: sj (pj , p j ) = exp ( β0 + screenj β1 + CPUj β2 + Weightj β3 αpj ) ∑j 0 exp ( β0 + screenj 0 β1 + CPUj 0 β2 + Weightj 0 β3 αpj 0 ) Qj = N
sj (pj , p j )
It can be veri…ed that the elasticity of the residual demand curve facing j is: %∆sj = αpj sj %∆pj
Patrick Bajari Econ 4631 ()
Oligopoly Models
38 / 55
Bertrand Equilibrium A Bertrand equilibrium is a Nash equilibrium in prices The pro…t of j is:
(pj
cj )Nsj (pj , p j )
The …rst order condtions are: ∂sj sj + (pj cj ) ∂pj
= 0
(pj
cj ) =
(pj
cj ) pj
= =
∆pj ∆sj sj ∆pj = pj ∆sj 1 εj
sj
%∆pj %∆sj
Where εj is the residual demand elasticity for good j Patrick Bajari Econ 4631 ()
Oligopoly Models
39 / 55
Measuring Market Power
The formula below is called the inverse elasticity rule:
(pj
cj ) pj
=
1 εj
This rule states that price cost margins are higher (and market power generates more ine¢ ciency), if the residual demand is more inelastic IO economists have been obsessed with the measurement of market power For example, they will go to the data and attempt to estimate εj
Patrick Bajari Econ 4631 ()
Oligopoly Models
40 / 55
Measuring Market Power
In the logit model, for example, εj =
αsj pj
If we know α, we can infer the Lerner index This is useful in practice since we can identify the most distorted markets Also, we can perhaps identify markets in which mergers might be particularly bad or compute the e¤ects of collusion This all boils down to determining how mergers or collusion changed residual demand elasticities
Patrick Bajari Econ 4631 ()
Oligopoly Models
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Calibration of the logit model
It is possible for us to calibrate the logit model in order to ask some interesting economic questions As an example, let’s consider the e¤ects of the proposed Google Yahoo! ad deal that was blocked by the Department of Justice Google- 65% Bing-10% Yahoo!-25%
Patrick Bajari Econ 4631 ()
Oligopoly Models
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Calibration of the logit model
Search engines make money by selling advertisements Advertisers bid for "sponsored links" on these sites They pay their bid times each click that is generated by a search
Patrick Bajari Econ 4631 ()
Oligopoly Models
43 / 55
Calibration of the logit model
Suppose that the utility of search engine j takes the form: uij = δj
αpj + εij
Where δj is the "mean" utility that an advertiser has for search engine j pj is the advertising price of search engine j εij is the preference shock for engine j
Patrick Bajari Econ 4631 ()
Oligopoly Models
44 / 55
Calibration of the logit model To simplify things, let’s suppose that prices of a click are equalized across sites Normalize prices to $1. Assume that Google’s price cost margin is 27% (this is from accounting data, not economic costs) This implies that:
p
c p
=
.27 = α =
Patrick Bajari Econ 4631 ()
1 αsj pj 1 α.65 1 = 5.6980 .27 .65 Oligopoly Models
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Calibration of the logit model
Next, let’s solve for the δj sGoog sBing sYahoo D
exp(δGoog 5.6980) D exp(δBing 5.6980) = .1 = D exp(δYahoo 5.6980) = .25 = D = exp(δGoog 5.6980) + exp(δBing
= .65 =
+ exp(δYahoo
5.6980)
5.6980)
Since we know all the shares, this is 3 equations in 3 unknowns
Patrick Bajari Econ 4631 ()
Oligopoly Models
46 / 55
Calibration of the logit model
Without loss of generality, we can normalize δBing = 0 Verify this by adding a constant to all of the δj Then we can solve for δGoog and δYahoo by: sGoog sBing δGoog δYahoo
Patrick Bajari Econ 4631 ()
.65 exp(δGoog 5.6980) .1 exp( 5.6980) = log(.65) log(.1) = 1.8718
=
= log(.25)
Oligopoly Models
log(.1) = .9163
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Post Merger Outcome
If we remove Yahoo! from the market, the shares of Bing and Google are: sGoog
=
sGoog
=
exp(1.8718 5.6980pGoog ) exp(1.8718 5.6980pGoog ) + exp( 5.6980pBing ) exp( 5.6980pBing ) exp(1.8718 5.6980pGoog ) + exp( 5.6980pBing )
Let’s give both Bing and Google costs of .83 cents (let’s ad in 10 cents of "economic costs" to the price cost margins)
Patrick Bajari Econ 4631 ()
Oligopoly Models
48 / 55
Post Merger Outcome
In a Nash equilibrium, the …rms would maximize:
(pGoog (pBing
exp(1.8718 5.6980pGoog ) exp(1.8718 5.6980pGoog ) + exp( 5.6980pBing ) exp( 5.6980pBing ) .83) exp(1.8718 5.6980pGoog ) + exp( 5.6980pBing ) .83)
Patrick Bajari Econ 4631 ()
Oligopoly Models
49 / 55
Post Merger Outcome
The Nash equilibrium would be the following two equations in the two unknowns pGoog , pBing
(pGoog .83) 1 = exp ( 1.8718 5.6980p ) pGoog 5.6980 exp (1.8718 5.6980p )+expGoog ( 5. Goog (pBing .83) pBing
=
1 exp ( 5.6980p Bing ) 5.6980 exp (1.8718 5.6980p )+exp p ( 5.6980p Bing ) Bing Goog
This can’t be solved for analytically. Instead, we do it numerically on the computer
Patrick Bajari Econ 4631 ()
Oligopoly Models
50 / 55
Calibration of the logit model
The post merger prices are pGoog = 1.4509, pBing = 1.2091 This is a very large price increase relative to the baseline prices of 1 This back of the envelope calculation suggests this acquisition could have been very harmful to advertisers
Patrick Bajari Econ 4631 ()
Oligopoly Models
51 / 55
Extensions of the Logit
Remark: What happens after this point of the lecture is a bit advanced and will not be on the exam While the logit model is widely used, there has been important work on extending this model In particular, researchers have found it important to include "random coe¢ cients in the logit" In the logit, the only reason why di¤erent people make di¤erent choices is because they draw di¤erent εij In the random coe¢ cient model, di¤erent people have di¤erent marginal utilities In practice, this turns out to be a very important extension
Patrick Bajari Econ 4631 ()
Oligopoly Models
52 / 55
Extensions of the Logit You will not be tested on this section One formulation of this model is to suppose that there are r = 1, ..., R types of consumers These types have utilities characterized by the parameters β(r ) , α(r ) This leads to the utility function (r )
(r )
(r )
(r )
uij = β0 + screenj β1 + CPUj β2 + Weightj β3
α(r ) pj + εij
Households are assumed to be utility maximizers, e.g. i chooses j if and only if uij > uij 0 for all j 0 6= i The probability that type r chooses good j is:
Patrick Bajari Econ 4631 ()
Oligopoly Models
53 / 55
Extensions of the Logit The type r’s choice probability is:
exp
(r ) β0
Pr(i chooses j jr ) = (r )
(r )
α(r ) pj
+ screenj β1 + CPUj β2
(r )
(r )
(r )
∑j 0 exp β0 + screenj 0 β1 + CPUj 0 β2
α(r ) pj 0
Let p (r ) denote the probability of type r The choice probability for j is found by summing over r
Pr( j chosen) = (r )
R
∑
r =1
p (r )
(r )
(r )
exp β0 + screenj β1 + CPUj β2 (r )
(r )
(r )
∑j 0 exp β0 + screenj 0 β1 + CPUj 0 β2
Patrick Bajari Econ 4631 ()
Oligopoly Models
α(r ) pj α(r ) pj 0 54 / 55
Extensions of the Logit
Next lecture, we will use this framework to study music piracy
Patrick Bajari Econ 4631 ()
Oligopoly Models
55 / 55
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